Tom Diethe, Department of Computer Science, University College London, UK
Zakria Hussain, Department of Computer Science, University College London, UK

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Abstract

Polytope Faces Pursuit (PFP) is a greedy algorithm that approximates the sparse solutions recovered by 1 regularised least-squares (Lasso) [4, 10] in a similar vein to (Orthogonal) Matching Pursuit (OMP) [16]. The algorithm is based on the geometry of the polar polytope where at each step a basis function is chosen by finding the maximal vertex using a path-following method. The algorithmic complexity is of a similar order to OMP whilst being able to solve problems known to be hard for (O)MP. Matching Pursuit was extended to build kernel-based solutions to machine learning problems, resulting in the sparse regression algorithm, Kernel Matching Pursuit (KMP) [17]. We develop a new algorithm to build sparse kernel-based solutions using PFP, which we call Kernel Polytope Faces Pursuit (KPFP). We show the usefulness of this algorithm by providing a generalisation error bound [7] that takes into account a natural regression loss and experimental results on several benchmark datasets.